1. Formal methods in software development
a.y.2016/2017
Prof. Anna Labella
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2. see dens.pdf
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10.  fi()  g Proof  f()  f2()  f3()…..........  fi()
f ( fi() =  fi+1()
Given another fixpoint g, we have the constant chain
 g  g  g…  g = g
greater than the first one step by step, hence
 fi()  g
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11. We could start from the maximum and compute the
maximal fixpoint using the duality of order
We can use minimum/ maximum fixed point to solve
recursive equations like
Ex: x=ax+b
If we do not have inverse operations, we have to
use approximation.
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12. Fixed point theorem: an example
A fixed point x for a function f:(S)(S) is an element of (S) such that f(x) = x
We will give an interpretation of CTL operators using fixed points
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13. Fixed point theorem: an example
Let S be a set (of states) and
f:(S)(S) a monotonic function w.r.t.  , then f has a minimal and a maximal fixpoint
nfn(Ø) and nfn(S), respectively
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14. Fixed point theorem (Tarski)
Let S be a set (of states) and
f:(S)(S) a monotonic function w.r.t.  ,
The maximal fixpoint of f S
f(S)
f2(S)
nfn(S)
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15. Looking for a semantic for CTL
i.e. find the set of states satisfying a formula.
This will be done through the notion of minimal/maximal
fixpoint of an operator.
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16. Recursively defined operators
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26. Fixed point theorem (Tarski)
We want to define the set of states where EGp is the greatest solution of the recursive equation x = p∧ EX (x)
i.e. as the maximal fixpoint of the operator p∧ EX : (S)(S)
We start from the greatest subset S
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